Schatten norm

In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm.

Definition

Let $H_{1}$ , $H_{2}$ be separable Hilbert spaces, and $T$ a (linear) bounded operator from $H_{1}$ to $H_{2}$ . For $p\in [1,\infty)$ , define the Schatten p-norm of $T$ as

\|T\| _{p} := \bigg( \sum _{n\ge 1} s^p_n(T)\bigg)^{1/p}

for $s_1(T) \ge s_2(T) \ge \cdots s_n(T) \ge \cdots \ge 0$ the singular values of $T$ , i.e. the eigenvalues of the Hermitian operator $|T|:=\sqrt{(T^*T)}$ . From functional calculus on the positive operator $T^{*}T$ it follows that

\|T\|_{{p}}^{p}={\mathrm {tr}}(|T|^{p}).

Properties

In the following we formally extend the range of $p$ to $[1,\infty ]$ . The dual index to $p=\infty$ is then $q=1$ .

The Schatten norms are isometrically invariant: for isometries $U$ and $V$ and $p\in [1,\infty ]$ ,

\|UTV\|_{p}=\|T\|_{p}.

They satisfy Hölder's inequality: for all $p\in [1,\infty ]$ and $q$ such that ${\frac {1}{p}}+{\frac {1}{q}}=1$ , and operators $S\in {\mathcal {L}}(H_{2},H_{3}),T\in {\mathcal {L}}(H_{1},H_{2})$ defined between Hilbert spaces $H_{1},H_{2},$ and $H_{3}$ respectively,

\|ST\|_{1}\leq \|S\|_{p}\|T\|_{q}.

Sub-multiplicativity: For all $p\in [1,\infty ]$ and operators $S\in {\mathcal {L}}(H_{2},H_{3}),T\in {\mathcal {L}}(H_{1},H_{2})$ defined between Hilbert spaces $H_{1},H_{2},$ and $H_{3}$ respectively,

\|ST\|_{p}\leq \|S\|_{p}\|T\|_{p}.

Monotonicity: For $1\leq p\leq p'\leq \infty$ ,

\|T\|_{1}\geq \|T\|_{p}\geq \|T\|_{{p'}}\geq \|T\|_{\infty }.

Duality: Let $H_{1},H_{2}$ be finite-dimensional Hilbert spaces, $p\in [1,\infty ]$ and $q$ such that ${\frac {1}{p}}+{\frac {1}{q}}=1$ , then

\|S\|_{p}=\sup \lbrace |\langle S,T\rangle |\mid \|T\|_{q}=1\rbrace ,

where $\langle S,T\rangle ={\mathrm {tr}}(S^{*}T)$ denotes the Hilbert-Schmidt inner product.

Remarks

Notice that $\|\cdot \|_{2}$ is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), $\|\cdot \|_{1}$ is the trace class norm (see trace class), and $\|\cdot \|_{\infty }$ is the operator norm (see operator norm).

For $p\in (0,1)$ the function $\|\cdot \|_{p}$ is an example of a quasinorm.

An operator which has a finite Schatten norm is called a Schatten class operator and the space of such operators is denoted by $S_p(H_1,H_2)$ . With this norm, $S_p(H_1,H_2)$ is a Banach space, and a Hilbert space for p = 2.

Observe that $S_p(H_1,H_2) \subseteq \mathcal{K} (H_1,H_2)$ , the algebra of compact operators. This follows from the fact that if the sum is finite the spectrum will be finite or countable with the origin as limit point, and hence a compact operator (see compact operator on Hilbert space).

References

Rajendra Bhatia, Matrix analysis, Vol. 169. Springer Science & Business Media, 1997.
John Watrous, Theory of Quantum Information, 2.3 Norms of operators, lecture notes, University of Waterloo, 2011.
Joachim Weidmann, Linear operators in Hilbert spaces, Vol. 20. Springer, New York, 1980.

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